Boundary Element Analysis Of Strain Gradient Elasticity Problems

Generalized continuum mechanics is different from classical continuum mechanics.It can analyze micro-scale problems from a macroscopic perspective and better explain the phenomenon of size effects.So far,many theoretical branches have been formed and developed.Since the strain gradient theory fully considers the effect of the second derivative of displacement,both the second derivative of the antisymmetric part of the displacement gradient and the second derivative of the symmetric part are considered,more research and applications have been obtained in recent years.The mature strain gradient elasticity theory explains many size effect phenomena,but when it comes to problems such as irregular geometric shapes and complex load environments,the limitations of analytical methods are obvious,so numerical techniques must be used.Boundary element method has many advantages such as reduced dimensionality of the problem,semi-analytical,high precision,suitable for dealing with large changes in field variable gradients,and no special requirements for element continuity.It is conducive to dealing with strain gradient elastic theory,so the boundary element method is used to carry out related researches on strain gradient theory.The main contents of this dissertation are:(1)The linear elastic displacement integral equation is established by using the reciprocal integral theorem of the strain gradient elastic theory,and the basic solution of the tensor form based on the strain gradient theory of Aifantis is expanded into the component form required by the program.(2)Since the boundary vector field included in the displacement boundary integral equation has displacement,displacement normal deviation,surface tractions and surface double stresses,it is necessary to add more boundary integral equations to obtain the solution of all unknown variables.Considering adding the displacement method to the number boundary integral equation to form the double boundary integral equation method.This method is used to simulate the gradient elastic bar,and the accuracy of the method is verified by comparison with the analytical solution.This method is also used to further analyze the force problems of cantilever beams,cylinders,and plate with elliptical hole.(3)Since the number of unknown variables of the displacement boundary integral equation is two more than that of the classical theory,considering the use of the relationship between the normal derivative of displacement and surface tractions and the displacement,using the variable substitution of the unknown quantity,thereby the unknown variables are reduced,forming a single boundary integral equation method.Using this method,the numerical analysis results satisfying the analytical solution of the classical elastic theory are obtained under special stress conditions.The calculation results of the two methods for the gradient elastic bar problem under different material characteristic lengths are compared,and the cylinder and plate with circle hole are further analyzed and processed.(4)Based on the strain gradient elastic theory,the Knein-Williams asymptotic expansion technique is used to derive the expansion formula of the plane crack tip displacement field and the relationship between the amplitude coefficients,and construct the solution format of the crack problem based on the strain gradient elastic theory.The analytical crack tip element technology is applied to the crack analysis of the classical continuum theory to verify the method,which provides a basis for further crack analysis of the theory.The programs of all the calculation examples in this dissertation are compiled and realized on the FORTRAN platform.